stationary waves
TRANSCRIPT
Stationary Waves
Stationary waves are produced by superposition of two progressive waves of equal amplitude and frequency, travelling with the same speed in opposite directions.
http://www2.biglobe.ne.jp/~norimari/science/JavaEd/e-wave4.html
Production of Stationary Waves
A stationary wave would be set up by causing the string to oscillate rapidly at a particular frequency.If the signal frequency is increased further, overtone patterns appear.
Properties of a stationary wave (1)
Stationary waves have nodes where there is no displacement at any time.In between the nodes are positions called antinodes, where the displacement has maximum amplitude.
A vibrating loop
N A N A N
VibratorVibrator
Properties of a stationary wave (2)
The waveform in a stationary wave does not move through medium; energy is not carried away from the source.The amplitude of a stationary wave varies from zero at a node to maximum at an antinode, and depends on position along the wave.
Vibrations of particles in a stationary waveAt t = 0, all particles are at rest because the particles reach their maximum displacements.At t = ¼T, Particles a, e, and i are at rest because they are the nodes.Particles b, c and d are moving downward. They vibrate in phase but with different amplitude.Particles f, g and h are moving upward. They vibrate in phase but with different amplitude.
t = 0
t = ¼T
t = ¼T
t = ⅜T
t = ½T
a
bc
d
ef
g
hii
ab c d
ef g h
ii
Properties of a stationary wave (2)
All particles between two adjacent nodes (within one vibrating loop) are in phase.
Video
1. Stationary waves (string)
2. Stationary waves (sound)
Standing waves in a string fixed at both ends.
Normal modes of a string
... 3, 2, 1,n 2
nL
n
... 3, 2, 1,n 2
Lvnvf
nn
Wavelength:
Frequency:
1
... 3, 2, 1,n T2
fnfLnf
n
n
Tv :Using
frequency lfundamenta thecalled is T21
1 Lf
Standing waves in a string fixed at both ends.
f1 is called the fundamental frequency
The higher frequencies fn are integer
multiples of the fundamental frequency
These normal modes are called harmonics.
f1 is the first harmonic, f2 is the second
harmonic and so on…
Modes of vibration of strings
Picture of Standing Wave Name Structure
1st Harmonic
orFundamental
1 Antinode2 Nodes
2nd Harmonic
or1st Overtone
2 Antinodes3 Nodes
3rd Harmonic
or2nd Overtone
3 Antinodes4 Nodes
4th Harmonic
or3rd Overtone
4 Antinodes5 Nodes
5th Harmonic
or4th Overtone
5 Antinodes6 Nodes
L = ½λ1
f1 = v/2LL = λ2
f2 = v/LL = 1½λ3
f3 = 3v/2LL = 2λ4
f4 = 2v/L
L = 2½λ5
f5 = 5v/2L
L
Investigating stationary waves using sound waves and
microwavesMoving the detector along the line between the wave source and the reflector enables alternating points of high and low signal intensity to be found. These are the antinodes and nodes of the stationary waves.The distance between successive nodes or antinodes can be measured, and corresponds to half the wavelength λ.If the frequency f of the source is known, the speed of the two progressive waves which produce the stationary wave can be obtained.
Reflector
DetectorWave source
Resonant Frequencies of a Vibrating String
From the experiment, we find that There is a number of resonant frequencies in a vibrating string,
The lowest resonant frequency is called the fundamental frequency (1st harmonic),
The other frequencies are called overtones (2nd harmonic, 3rd harmonic etc.),
Each of the overtones has a frequency which is a whole-number multiple of the frequency of the fundamental.
Factors that determine the fundamental frequency of a
vibrating string
The frequency of vibration depends on the mass per unit length of the string, the tension in the string and, the length of the string.The fundamental frequency is given by
T
Lfo 2
1 where T = tension
= mass per unit length L = length of string
Vibrations in Air Column
When a loudspeaker producing sound is placed near the end of a hollow tube, the tube resonates with sound at certain frequencies.Stationary waves are set up inside the tube because of the superposition of the incident wave and the reflected wave travelling in opposite directions.
http://www.walter-fendt.de/ph11e/stlwaves.htm
Factors that determine the fundamental frequency of a
vibrating air columnThe natural frequency of a wind instrument is dependent upon The type of the air column, The length of the air column of the instrument.
Open tube Closed tube
Standing Waves in a String, 3This is the first normal mode
that is consistent with the boundary conditions.There are nodes at both ends.There is one antinode in the
middle.This is the longest
wavelength mode: ½ = L so = 2L
The section of the standing wave between nodes is called a loop.In the first normal mode, the
string vibrates in one loop.
Section 18.3
Standing Waves in a String, 4
Consecutive normal modes add a loop at each step. The section of the standing wave from one node to the next is called
a loop.The second mode (c) corresponds to to = L.The third mode (d) corresponds to = 2L/3.
Section 18.3
/4/4
Name
Modes of vibration for an open tube
Picture of Standing Wave Structure
1st Harmonic
orFundamental
2 Antinodes1 Node
2nd Harmonic
or1st Overtone
3 Antinodes2 Nodes
3rd Harmonic
or2nd Overtone
4 Antinodes3 Nodes
4th Harmonic
or3rd Overtone
5 Antinodes4 Nodes
5th Harmonic
or4th Overtone
6 Antinodes5 Nodes
L = ½λ1
f1 = v/2L
L = λ2
f2 = v/L
L = 1½λ3
f3 = 3v/2L
L = 2λ4
f4 =2v/LL = 2½λ5
f5 = 5v/2L
Modes of vibration for a closed tube
Picture of Standing Wave Name Structure
1st Harmonic
orFundamental
1 Antinode1 Node
3rd Harmonic
or1st Overtone
2 Antinodes2 Nodes
5th Harmonic
or2nd Overtone
3 Antinodes3 Nodes
7th Harmonic
or3rd Overtone
4 Antinodes4 Nodes
9th Harmonic
or4th Overtone
5 Antinodes5 Nodes
L = ¼λ1
f1 = v/4L
L = ¾λ3
f3 =3v/4L
L = 1¼λ5
f5 =5v/4L
L = 1¾λ7
f7 = 7v/4L
L = 2¼λ9
f9 =9v/4L
The quality of sound (Timbre)
The quality of sound is determined by the following factors: The particular harmonics present in addition to the fundamental vibration,
The relative amplitude of each harmonic, The transient sounds produced when the vibration is started.
1st overtone Fundamental2nd overtone
3rd overtone
resultant
http://surendranath.tripod.com/Harmonics/Harmonics.html
Chladni’s PlateChladni’s plate is an example of resonance in a plate.There are a number of frequencies at which the plate resonate. Each gives a different pattern.
Resonance. All elastic objects have natural frequencies of vibration that are determined by the materials they are made of and their shapes.
When energy is transferred at the natural frequencies, there is a dramatic increase of amplitude called resonance.
The natural frequencies are also called resonant frequencies.
When the frequency of an applied force, including the force of a sound wave, matches the natural frequency of an object, energy is transferred very efficiently. The condition is called resonance.